- Your 2.4 homework is
returned: graded #6, 36, and 42.
Two different approaches to #42 (I love mathematics because there's always another way to do the problem....).
Though you may be tempted, Don't use L'Hopital's rule yet (in case you know what that means), unless you can prove it.
Notice that in problem #36, $a(t)=s''(t)=-s(t)$: this is the differential equation whose solutions are sines and cosines.
To draw sines and cosines, units of $\pi$ along the $x$-axis are best. You're not stuck with integers, you know: you can get all irrational.
- Please put your rectangles away.
- In calculus you encounter the famous ladder problem. Example 2 on
p. 177. We wish to establish the rate of one thing, relative to the
rate of the other. These are so-called "related rates". So the ladder's
top rung is falling vertically at one speed, as the feet are slipping
out at another. Speeds are, in this case, rates of change of positions.
- The problems considered here are the fun ones, the story problems:
- Everyone loves story problems, right?! Let's go over the strategy,
and then try some.
I jokingly use the acronym "UPCE" (oopsie!) for the general problem solving strategy:
- Understand
- Plan
- Carry out
- Evaluate
- Here is our author's more detailed approach to these related rates
problems:
- Read the problem carefully.
- Draw a diagram if possible.
- Introduce (good) notation. Use sensible variable
names. Assign symbols to all quantities that are
functions of time (usually time will be our
independent variable).
- Express the given information and the required rate in
terms of derivatives.
- Write an equation that relates the various quantities of
the problem. If necessary, use the geometry of the situation to
eliminate one of the variables by substitution (as in Example 3).
- Use the Chain Rule to differentiate both sides of the
equation with respect to t.
- Substitute the given information into the resulting
equation and solve for the unknown rate.
- Don't forget your units.
"Warning: a common error is to substitute the given numerical information (for quantities that vary with time) too early." (p. 179). Substitute only after the differentiation is complete.
- Read the problem carefully.
- Examples:
- #4, p. 180 (basic example, no chain rule yet!).
- #6 (yep -- chain rule!)
- #23 (Key: similar triangles -- but this is a challenging one!)
- #31 (ladders! Pythagoras....)
- #34 (straightforward, implicit differentiation with chain rule)
- #37 (law of cosines)
- #38 looks like fun....